softies can just go in a bag of water, for certain lps and sps, put them on some polystyrene using an elastic band and put them in the water upside down. This will ensure they are always submerged. Pack them in a box with a heat pack and plenty of bubble wrap. Should be ok for a while like that.
How far away are you moving, I have moved several aquariums with great success. But I pack and move them differently depending on how far I'm moving them.
For a move across town, just get yourself a styrofoam cooler, put some water in it, and toss the corals in (gently). There's no need to throw the corals out when you move, they are pretty easy to transport. When you buy corals online, they are shipped overnight. They will do just fine in a styrofoam cooler or plastic bags for 24+ hours.